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The second assumption we will make is that the Lagrangian fibration is principally po- larized, in the sense explained in Definition 3.1. This condition is satisfied if there exists 1 an f-ample line bundle on M f − (D) whose restriction on smooth fibers give principal polarizations on the abelian varieties.\ This assumption is rather restrictive compared with the setting of [HO1] and [HO2], where the only assumption was that the fibers of f are of Fujiki class. We believe that understanding the structure of Lagrangian fibration under these as- sumptions is essential for the study of general cases. Since this special case already requires substantial care and already provides many interesting examples (see Section 5), we restrict our attention to it in this paper and leave the general cases to future study. The main result of this paper is the following.
Theorem 1.1. Let f : M B be a principally polarized stable Lagrangian fibration (cf. Definition 2.1 and Definition→ 3.1). Then at a general point b D of the discriminant, there ∈ exists a coordinate system (z1,...,zn) with D defined by zn = 0, such that for a suitable choice of an integral frame of the local system R1f Z on B D, the period matrices have the form ∗ \
2 2 i ∂ Ψ n ∂ Ψ ℓ θj = for (i, j) = (n,n) and θn = + log zn ∂zi∂zj 6 ∂zn∂zn 2π√ 1 − where Ψ is a holomorphic function in z1,...,zn. Here ℓ is the number of irreducible com- ponents of a general singular fiber. i Conversely, given any germ of holomorphic function Ψ(z1,...,zn) such that Im(θj) > 0, there exists a principally polarized stable Lagrangian fibration whose period matrices are of the above form.
That the period matrix of Lagrangian fibration is the Hessian of a potential function is a well-known consequence of the action-angle variables (cf. [DM]). The logarithmic behavior of the multi-valued part reflects the stability assumption on the singular fiber. The novelty in Theorem 1.1 lies in the choice of the variable zn through which these two aspects are intertwined. The existence of zn follows from the fact proved in Proposition 3.13 that the characteristic foliation accounts for the degenerate part of the polarization restricted to the fixed part of the monodromy. The proof of this uses a version of the Monodromy Theorem from the theory of the degeneration of Hodge structures and the topological property of the stable singular fiber. The converse direction in Theorem 1.1 is shown by explicitly constructing a principally polarized stable Lagrangian fibration from a given potential function Ψ(z) imaginary part of whose Hessian matrix is positive definite. This part is a sort of generalization of Nakamura’s construction [Na] of toroidal degeneration of principally polarized abelian varieties over 1- dimensional small disk. Using our construction, we shall give a concrete 4-dimensional example of principally polarized stable Lagrangian fibration in which the types of char- acteristic cycles of singular fibers change fiber by fiber, too. To our knowledge, such an example has not been noticed previously. In fact, most of the previous constructions of singular fibers of Lagrangian fibrations have used product construction from elliptic fibra- tions. LOCAL STRUCTURE OF PRINCIPALLY POLARIZED STABLE LAGRANGIAN FIBRATIONS 3
2. Stable Lagrangian fibrations Definition 2.1. A Lagrangian fibration is a proper flat morphism f : M B from a holomorphic symplectic manifold (M,ω) of dimension 2n to a complex manifold→ B of di- mension n such that the smooth locus of each fiber is a Lagrangian submanifold of M. The discriminant D B is the set of the critical values of f, which is a hypersurface in B if it is non-empty. Throughout⊂ this paper, we assume that D is non-empty. We say that f is a 1 stable Lagrangian fibration if D B is a submanifold and each singular fiber f − (b), b D, is stable, i.e., it is reduced and⊂ the characteristic cycle in the sense of [HO1] is of∈ type 1 Ik, 1 k . By the description in [HO1], this is equivalent to saying that f − (b) is reduced≤ and≤ ∞its normalization is a disjoint union of a finite number of compact complex manifolds Y 1,...,Y ℓ for some positive integer ℓ such that (i) each Y i is a P1-bundle over an (n 1)-dimensional complex torus Ai whose fibers − 1 are sent to characteristic leaves of f − (b) in the sense of [HO1], i.e., for a defining function h (B) of the divisor D, the Hamiltonian vector field ιω(f ∗dh), where 1 ∈ O ιω :ΩM T (M) is the vector bundle isomorphism induced by ω, is tangent to the image of→ the fibers in M; i i i i i (ii) there exist submanifolds S1,S2 Y , with S1 = S2 except possibly when ℓ = 1, 2, i i ⊂ 6 i 1 such that S1 S2 is a 2-to-1 unramified cover of A under the P -bundle projection; ∪ i 1 (iii) the normalization ν : Y f − (b) is obtained by the identification via a collection → i i+1 ℓ 1 of biholomorphic morphismsS gi : S2 S1 for 1 i ℓ 1 and gℓ : S2 S1 with 1→ 1 1 ≤ ≤2 − 2 → the additional requirement g1 = g2− if S1 = S2 and S1 = S2 for ℓ = 2. A maximal connected union of the P1-fibers in (i) under the identification in (iii) is called a characteristic cycle. A characteristic cycle can be either of finite type (Im-type, 1 m< ) or of infinite type A , which we also denote by I . ≤Recall∞ (cf. [HO2] Section 4) that∞ in a neighborhood of a gener∞ al singular fiber, any La- grangian fibration whose fibers are of Fujiki class can be transformed to a stable Lagrangian fibration by certain explicitly given bimeromorphic modifications and branched covering. We will be interested in the local property of the fibration at a point of D. Thus we will make the following (Assumption) D B is the germ of a smooth hypersurface in an n-dimensional complex manifold and the fundamental⊂ group π (B D) is cyclic. 1 \ The following is immediate from Proposition 2.2 of [HO1].
Proposition 2.2. Given a stable Lagrangian fibration, we can assume that there exists an n 1 action of the complex Lie group C − on M preserving the fibers and the symplectic form i i n 1 i such that S1,S2 are orbits of this action for all 1 i ℓ. This action of C − on Y i ≤ ≤ descends to the translation action on A . The patching biholomorphisms gi in Definition 2.1 (iii) as well as the P1-bundle structure in (i) are equivariant under this action. In 1 1 2 2 1 1 particular, if S1 = S2 (resp. S1 = S2 ), the Galois action of the double cover S1 A (resp. S2 A2) is by a translation on the torus S1 (resp. S2). → 1 → 1 1 Regarding the topology of the singular fiber f 1(b), b D, we have the following. − ∈ 4 JUN-MUK HWANG, KEIJI OGUISO
Proposition 2.3. In the setting of Definition 2.1, fix a component Y 1 of the normalization 1 of f − (b) and set Y := Y 1 (S1 S1), o \ 1 ∪ 2 which is equipped with a C∗-bundle structure ̺ : Yo A over a complex torus A of dimen- sion n 1 coming from Definition 2.1 (i). Then there→ exists a (not necessarily holomorphic) − 1 continuous map µ : f − (b) A′ to a complex torus A′ isogenous to A such that when 1 → j : Y0 f − (b) is the natural inclusion and ρ : Y0 A A′ is the composition of ̺ and an isogeny,→ µ j is homotopic to ρ. → → ◦ Proof. Let us use the notation introduced in Definition 2.1 (iii) for the description of the normalization morphism ν : Y i f 1(b). → − Si i ^1 First, we consider the case S1 = S2 for all 1 i ℓ. Define f − (b) as the variety obtained i 6 ≤ ≤ from Y with all the patching identification g1,...,gℓ 1 such that the normalization − factorsS through i ^1 1 ν : Y f − (b) f − (b) [ → → ^1 1 with the second arrow given by the identification via gℓ. When ℓ = 1, f − (b) = Y . The n 1 n 1 ^1 C − -action of Proposition 2.2 lifts to a C − -action on f − (b). The connected unions of the images of the P1-fibers of Y i define finite chains of quasi-transversally intersecting 1 ^1 P ’s in f − (b), which we call characteristic chains. Each characteristic chain intersects ′′ i i ^1 each S1 (resp. S2), 1 i ℓ at exactly one point, inducing a morphism f − (b) A to a complex torus of dimension≤ ≤ n 1 biholomorphic to Ai’s. This determines a biholomorphism→ ζ : Sℓ S1. Fix a point α −Sℓ and let β = ζ(α) S1. For t [0, 1] R, let τ : S1 S1 2 → 1 ∈ 2 ∈ 1 ∈ ⊂ t 1 → 1 be the translation by t(β gℓ(α)). − t ℓ 1 t Define a new family of biholomorphic morphisms gℓ : S2 S1 by gℓ = τt gℓ. Clearly, 0 1 1 1 → ℓ ◦ gℓ = gℓ. We claim that gℓ = ζ. In fact, ζ− gℓ is an automorphism of S2 which fixes the 1 ◦ n 1 point α. But both gℓ and ζ must be equivariant under the C − -action of Proposition 2.2. Thus ζ 1 g1 must be the identity map of Sℓ, proving the claim. − ◦ ℓ 2 1 t ^1 1 ℓ t Let f − (b) be the variety obtained from f − (b) by identifying S1 and S2 via gℓ. Then 1 0 1 f − (b) = f − (b) 1 1 1 n 1 1 t and f − (b) is homeomorphic to f − (b). The C − -action descends to f − (b) for each t as n 1 τt commutes with the C − -action. By abuse of terminology, we call the maximal connected 1 t 1 t unions of the images in f − (b) of the characteristic chains as characteristic cycles of f − (b) . n 1 1 t By the C − -action, we know that all characteristic cycles in f − (b) are isomorphic. By 1 1 our choice of α and β, there exists one finite characteristic cycle in f − (b) . Thus we 1 1 ′′ get a morphism µ′ : f − (b) A′ to some complex torus A′ isogenous to A . Define 1 → 1 1 1 µ : f − (b) A′ as the composition of µ′ with the homeomorphism f − (b) f − (b) . It certainly satisfies→ the required property. →
1 1 ^1 1 1 1 Now consider the case when ℓ = 1 and S1 = S2 . Set f − (b)= Y and define ζ : S2 S1 1 1 → as the Galois action of the double covering S1 A in Definition 2.1 (ii). Then the same argument as in the previous case applies. → LOCAL STRUCTURE OF PRINCIPALLY POLARIZED STABLE LAGRANGIAN FIBRATIONS 5
1 1 2 2 Finally, consider the case when ℓ = 2, S1 = S2 and S1 = S2 . By Proposition 2.2, 1 2 1 2 the Galois action on S1 (resp. S1 ) of the double cover over A (resp. A ) is given by n 1 n 1 1 a translation, say, by γ1 C − (resp. γ2 C − ). By the equivariance of g1 = g2− , 1 ∈ ∈ for each α S1 , we have g1(γ1 α) = γ2 g1(α). Thus by the normalization morphism 1 ∈2 1 · 1 · 2 ν : Y Y f − (b), a point α S1 is identified with g1(α) S1 , and the point ∪ 1 → 1 ∈ ∈ γ1 α S1 , which lies in the P -fiber through α, is identified with γ2 g1(α), which lies · ∈ 1 1 · in the P -fiber through g1(α). Thus we get a morphism µ : f − (b) A′ to an (n 1)- 1 → − dimensional torus A′ whose fiber is a union of two P ’s identified at two points. This µ satisfies the required property.
We have the generalization of the classical action-angle correspondence as follows. Proposition 2.4. Given a stable Lagrangian fibration f : M B, choose a Lagrangian → section Σ M of f. Then we have a natural surjective unramified morphism Φ : T ∗B M E where⊂ E is the union of the irreducible components of the fibers of f disjoint from→ Σ such\ that (1) f Φ agrees with the natural projection g : T B B, ◦ ∗ → (2) Φ sends the zero section of T ∗B to Σ and (3) Φ∗ω coincides with the standard symplectic form on T ∗B. 1 In particular, Γ:=Φ− (Σ) is a Lagrangian submanifold (with many connected components) in T B. For each b B, Φ := Φ ∗ : T (B) f 1(b) E is the universal covering and ∗ ∈ b |Tb (B) b∗ → − \ Γ := Γ T (B) is naturally isomorphic to H (f 1(b) E, Z). b ∩ b∗ 1 − \ Proof. Over B D, this is just a holomorphic version (cf. Proposition 3.5 in [Hw]) of the classical action-angle\ correspondence as described in Section 44 of [GS]. The statement over D follows by the same argument as for the smooth fibers. In fact, for each b B, the 1 ∈ vector group Tb∗(B) acts on the fiber f − (b) with n-dimensional orbits on the smooth locus 1 of f − (b) (cf. Proposition 3.3 in [Hw]). The morphism Φb is defined by taking the orbit 1 map of the point Σ f − (b) under this action, which is a universal covering map for the ∩ 1 1 smooth locus of the component of f − (b) containing Σ f − (b). This shows (1) and (2). The proof of (3) is the same as that of Theorem 44.2 of∩ [GS]. Proposition 2.5. Let b D. In the notation of Proposition 2.4, we can assume that the ∈ connected component of Γ containing each point of Γ Tb∗(B) is a Lagrangian section of T (B) B, i.e., a closed 1-form on B. Let Γ Γ be∩ the union of such sections of Γ over ∗ → ′ ⊂ B. Then for each s B D, Γ := Γ T (B) is a sublattice of Γ satisfying Γ /Γ = Z. ∈ \ s′ ′ ∩ s∗ s s s′ ∼ Proof. Since f 1(b) E is a C -bundle over an (n 1)-dimensional torus, we see that − \ ∗ − Γ Tb∗(B) has rank 2n 1. Thus Γs′ has rank 2n 1. It remains to show that Γs/Γs′ is torsion-free.∩ Suppose it− has a k-torsion, 0 < k Z−, i.e., there exists a point α Γ Γ ∈ ∈ s \ s′ such that kα Γ . Let kα be a closed 1-form given by the component of Γ containing kα. ∈ s′ ′ Then the closed 1-formα ˜ = 1 kα is also a component of Γ containing α, which implies f k ′ α Γ , a contradiction. ∈ s′ f Proposition 2.6. In the notation of Proposition 2.4, let Yo be the fiber of M E at a point b D. Let Φ : T (B) Y be the universal covering map and ̺ : Y A be the\ C -bundle ∈ b b∗ → o o → ∗ 6 JUN-MUK HWANG, KEIJI OGUISO over an (n 1)-dimensional torus. Let Υ Γb =Γb′ be the rank-1 sublattice corresponding to the kernel− of ⊂ ̺ : H1(Y0, Z) H1(A, Z). ∗ → 1 1 Then for any v Γb′ Υ, there exists ̟ H (f − (b), Z) such that ̟, j v = 0 where ∈ \1 ∈ 1h ∗ i 6 j : H1(Yo, Z) H1(f − (b), Z) is induced by the inclusion j : Yo f − (b). ∗ → ⊂ 1 Proof. Since ̺ (v) H1(A, Z) is non-zero, there exists ϕ H (A, Z) such that ϕ, ̺ (v) = ∗ ∈ 1 ∈ h ∗ i 6 0. Let ̟ = µ∗ϕ where the map µ : f − (b) A is as defined in Proposition 2.3 satisfying µ j = ̺. Then → ◦ ̟, j (v) = µ∗ϕ, j (v) = j∗µ∗ϕ, v = ̺∗ϕ, v = ϕ, ̺ (v) = 0. h ∗ i h ∗ i h i h i h ∗ i 6
For a stable Lagrangian fibration f : M B, denote by Λ the local system on B D defined by the lattice Λ := H (f 1(s), Z) for→s B D. \ s 1 − ∈ \ Proposition 2.7. For a stable Lagrangian fibration f : M B and s B D, fix a generator of the cyclic fundamental group of π (B D,s) and→ denote by τ ∈: Λ \ Λ the 1 \ s s → s monodromy operator of the generator. Then the fixed part Λs′ Λs of τs at s B D has corank 1. ⊂ ∈ \
Proof. For any s B D, we can identify each fiber Λs = H1(Ms, Z) with the fiber Γs of Proposition 2.4. Thus∈ \ the result follows from Proposition 2.5.
3. Principally polarized stable Lagrangian fibration Definition 3.1. Let Λ be as in Proposition 2.7. A principal polarization on a stable 2 Lagrangian fibration f : M B is a unimodular anti-symmetric form Q : Λ ZB D → ∧ → \ where ZB D denotes the constant sheaf of integers on B D, which induces a principal polarization\ on each smooth fibers of f. A stable Lagrangian\ fibration with a choice of principal polarization is called a principally polarized stable Lagrangian fibration. 1 Remark 3.2. In Definition 3.1, the polarization on M f − (D) may not extend to an f-ample class of the whole M. In fact, f need not be projective.\ This definition is useful because there are many situations where the polarization exists a priori only on the smooth fibers, e.g., in Kodaira’s study of elliptic fibrations and also in our construction in Section 5. Proposition 3.3. Let f : M B be a principally polarized stable Lagrangian fibration. Then the monodromy operator→ in Proposition 2.7 satisfies τ = Id and τ τ = Id. s 6 s ◦ s 6 Proof. If τs = Id, then we see that f is a smooth fibration, as in the proof of Proposition 3.2 in [Hw]. In fact, since there is no monodromy and f is polarized over B D, we can extend the period map of the abelian family on B D to the whole B ([Gr], Theorem\ 9.5). Thus, we obtain a smooth abelian fibration f : M\ B such that f and f are bimeromorphic ′ ′ → ′ outside D. Since M ′ contains no rational curves and both M and M ′ have trivial canonical bundles, this implies M and M ′ are biholomorphic, a contradiction to the non-emptiness of the discriminant D of f. LOCAL STRUCTURE OF PRINCIPALLY POLARIZED STABLE LAGRANGIAN FIBRATIONS 7
If τ τ = Id, take a double cover g : B B branched along D and let D = g 1(D). s ◦ s ′ → ′ − Denote by fˆ : Mˆ B′ the fiber product of f and g, which has no monodromy on B′ n 1 → \ D′. By the C − -action of Proposition 2.2 which lifts to Mˆ , the following property of Mˆ can be seen from the corresponding properties in the case of n = 1 (cf. Proof of Proposition 9.2 in [BHPV]): Mˆ is normal, Gorenstein with singularities of type A1 ( germ of (2n 1)-dimensional manifold ) and has trivial canonical bundle. Thus we have× − a crepant resolution f ′ : M ′ B′, which is a family with trivial canonical bundle and no monodromy. Then we get a contradiction→ as in the previous case. Lemma 3.4. Let τ : Λ Λ be an automorphism of a lattice such that Λ := v Λ,τ(v)= → ′ { ∈ v is a sublattice of corank 1, i.e., Λ/Λ = Z. If τ τ = Id, then η := τ Id satisfies η η = 0. } ′ ∼ ◦ 6 − ◦ Proof. Note that Λ′ Ker(η). The induced automrophismτ ¯ : Λ/Λ′ Λ/Λ′ is either Id or Id. Ifτ ¯ = Id, then⊂ for a non-zero v Λ Λ , we have τ(v) = v +→λ for some λ Λ . − ∈ \ ′ ∈ ′ Then η(v)= λ Λ′ Ker(η). This proves that η η = 0. Ifτ ¯ = Id, then for a non-zero v Λ Λ , we have∈ τ⊂(v)= v + λ for some λ Λ◦. Then − ∈ \ ′ − ∈ ′ τ τ(v)= τ(v)+ τ(λ)= ( v + λ)+ λ = v. ◦ − − − Thus τ τ = Id, a contradiction. ◦ Proposition 3.5. In the setting of Proposition 3.3, let η := τs Id. Then for any β Im(η) and an element ϕ H1(M, Z), − ∈ ∈ i∗ϕ, β = ϕ, i β = 0 h i h ∗ i 1 1 where i∗ : H (M, Z) H (Ms, Z) and i : H1(Ms, Z) H1(M, Z) are the homomorphisms induced by the inclusion→ i : M := f 1(s∗) M. → s − ⊂ 1 Proof. Let be the local system on B D given by H (Ms, Z),t B D. Denote by τ : H the transformation dual to τ\, i.e., for any ̟ H1(M , Z∈) and\u H (M , Z), ∗ Hs →Hs ∈ s ∈ 1 s τ ∗(̟), u = ̟,τ(u) . h i h i By Proposition 2.7, Proposition 3.3 and Lemma 3.4, we have η = 0 and η η = 0, i.e., 6 ◦ 0 = Im(η) Ker(η) = Λ′ . 6 ⊂ s Similarly, η := τ Id is an endomorphism of with η = 0 and η η = 0. Since ∗ ∗ − Hs ∗ 6 ∗ ◦ ∗ i∗ϕ Ker(η∗) by (the easy half of) the global invariant cycles theorem (cf. Theorem 4.24 of [Vo]),∈ for any ψ Ker(η ) and u Λ , ∈ ∗ ∈ s ψ, η(u) = η∗(ψ), u = 0. h i h i It follows that i ϕ, Im(η) = 0. h ∗ i Remark 3.6. If the family f : M B is projective, we could have used the Monodromy Theorem (cf. Theorem 3.15 in [Vo])→ in place of Proposition 3.3 and Lemma 3.4 in the above proof. We have used the above approach because we do not want to assume that f is projective. Proposition 3.7. For a principally polarized stable Lagrangian fibration f : M B and s B D, let τ : Λ Λ be the monodromy operator of Proposition 2.7, which→ should ∈ \ s s → s 8 JUN-MUK HWANG, KEIJI OGUISO
2 preserve the polarization Qs : Λs Z. Setting η := τs Id as in Proposition 3.5, we have Λ = Ker(η). Then Im(η)∧ Λ →is contained in − s′ ⊂ s Ξ := v Λ′ Q(v, w) = 0 for all w Λ′ . s { ∈ s | ∈ s} Proof. Since τ preserves the polarization Q and η η = 0 by Lemma 3.4, s s ◦ Q (η(v), u)+ Q (v, η(u)) = 0 for all v, u Λ . s s ∈ s Thus for any v Λs and u Ker(η) = Λs′ , we have Qs(η(v), u) = Qs(v, η(u)) = 0, which means η(v) Ξ∈. ∈ − ∈ s Definition 3.8. Let Λ be a free abelian group of rank 2n. Given a unimodular non- 2 degenerate anti-symmetric form Q : Λ Z, a basis p1,...,pn,q1,...,qn of Λ is called a symplectic basis of Λ with respect to∧Q if,→ in terms of the{ dual basis p1,...,p} n,q1,...,qn of Hom(Λ, Z), { } Q = p1 q1 + p2 q2 + + pn qn. ∧ ∧ · · · ∧ Lemma 3.9. In the setting of Definition 3.8, let τ : Λ Λ be a group automorphism pre- serving Q. Assume that the subgroup Λ Λ of elements→ fixed under τ has corank 1. Then ′ ⊂ there exists a symplectic basis p1,...,pn,q1,...,qn such that p1,...,pn,q1,...,qn 1 { } { − }⊂ Λ′. Proof. Fix a symplectic basis a , . . . , a , b , . . . , b such that { 1 n 1 n} Q = a1 b1 + + an bn. ∧ · · · ∧ The anti-symmetric form Q ′ must have a kernel of rank 1, i.e., |Λ Ξ := v Λ′, Q(v, u) = 0 for all u Λ′ { ∈ ∈ } has rank 1. Pick a generator p1 of Ξ. Since Ξ is primitive, i.e., Λ/Ξ has no torsion, we can write p = α a + + α a + β b + + β b 1 1 1 · · · n n 1 1 · · · n n with some integers αi, βi satisfying gcd(α1,...,αn, β1,...,βn) = 1. Thus there exists inte- gers α1′ ,...,αn′ , β1′ ,...,βn′ such that α′ α + + α′ α + β′ β + + β′ β = 1. 1 · 1 · · · n · n 1 · 1 · · · n · n Let q := β′ a β′ a + α′ b + + α′ b . 1 − 1 1 −···− n n 1 1 · · · n n Then Q(p1,q1) = 1. Define ′′ Λ := v Λ,Q(p1, v)=0= Q(q1, v) . ′′ { ∈ } Then Λ Λ is a lattice of rank 2n 2 such that Q ′′ is unimodular and non-degenerate ⊂ ′ − |Λ (cf. [GH], the proof of Lemma in p.304). Let p2,...,pn,q2,...,qn be a symplectic basis of ′′ { } Λ . Then p ,...,p ,q ,...,q is a symplectic basis of Λ with the required property. { 1 n 1 n} Proposition 3.10. In the setting of Proposition 2.5, identify Λs = H1(Ms, Z) with Γs for s B D as in the proof of Proposition 2.7. Assume that we have a principal polarization ∈ \ Q. Then we can find a collection of components p1,...,pn,q1,...,qn 1 of Γ′ such that for each x B D, there exists q Γ such that {p ,...,p ,q ,...,q− } is a symplectic ∈ \ n,x ∈ x { 1,x n,x 1,x n,x} basis of Λx =Γx with respect to Qx. LOCAL STRUCTURE OF PRINCIPALLY POLARIZED STABLE LAGRANGIAN FIBRATIONS 9
Proof. Fix a point s B D. The monodromy τ : Λ Λ preserves the polariza- ∈ \ s s → s tion Qs on Λs and fixes Λs′ = Γs′ . Applying Lemma 3.9, we have a symplectic basis p1,s,...,pn,s,q1,s,...,qn,s with p1,s,...,pn,s,q1,s,...,qn 1,s Λs′ . Since Γ′ consists of sec- { } − ∈ tions of g : T ∗B B, the vectors p1,s,...,pn,s,q1,s,...,qn 1,s uniquely determine compo- → − nents p1,...,pn,q1,...,qn 1 of Λ′. To check the existence of qn,x for any x B D, just − ∈ \ pick qn,x as any vector in Λx contained in the component of Λ containing qn,s. Proposition 3.11. In the notation of Proposition 3.10, when b D, the vector p Γ ∈ n,b ∈ b′ regarded as an element of H1(Yo, Z) in the notation of Proposition 2.6, lies in the lattice Υ of Proposition 2.6. Proof. Suppose not. By our (Assumption) after Definition 2.1, we may assume that M 1 1 1 1 is topologically retractable to f − (b) and identify H (f − (b), Z) with H (M, Z). Then by 1 1 1 Proposition 2.6, there exists ̟ H (f − (b), Z)= H (M, Z) such that ̟, j pn,b = 0. For ∈ h ∗ i 6 a point s B D, the choice in Proposition 3.10 implies that pn,s Ξs of Proposition 3.7. ∈ \ 1 1 ∈ Denote by ̟s the element in H (Ms, Z) induced by ̟ H (M, Z) under the identification 1 1 1 1 ∈ H (f − (b), Z) = H (M, Z). Since j pn,b H1(f − (b), Z) = H1(M, Z) and the image of ∗ ∈ pn,s H1(Ms, Z) in H1(M, Z) belongs to the same class, Proposition 3.5 and Proposition 3.7 say∈ that ̟, j pn,b = ̟s,pn,s = 0. h ∗ i h i This is a contradiction.
Proposition 3.12. In Proposition 3.11, the C-linear span of Υ in Tb∗(B) is exactly C dh where h (B) is a defining equation of the divisor D. · ∈ O Proof. From the definition of Υ in Proposition 2.6, the linear span of Υ is sent to a fiber of the C∗-bundle. By Definition 2.1 (i), this fiber is a leaf of the characteristic foliation, which is given by the Hamiltonian vector field ιω(f ∗dh) on M. Under the symplecto-morphism Φ in Proposition 2.4, this corresponds to C dh. · Proposition 3.13. Let p1,...,pn,q1,...,qn 1 be as in Proposition 3.10. Then there exists a holomorphic coordinate{ system z ,...,z− } on B such that, regarded as sections of { 1 n} T ∗(B), p1 = dz1, ..., pn = dzn and D is given by zn = 0.
Proof. Since p1,...,pn are closed 1-forms which are point-wise linearly independent at every point of B, we can find coordinates z1,...,zn with pi = dzi. By Proposition 3.12, we may choose zn to be a defining equation of D. Let us recall the classical Riemann condition (e.g. [GH], p.306). Proposition 3.14. Let V be a complex vector space of dimension n and let Λ V be a lattice of rank 2n such that V/Λ is an abelian variety with a principal polarization.⊂ For a symplectic basis p ,...,p ,q ,...,q of Λ with respect to the principal polarization { 1 n 1 n} Q : 2Λ Z, p ,...,p becomes a C-basis of V and the period matrix (θj) defined by ∧ → { 1 n} i n j qi = θi pj in V Xj=1 10 JUN-MUK HWANG, KEIJI OGUISO
j is symmetric in (i, j) and Im(θi ) > 0. Theorem 3.15. Given a principally polarized stable Lagrangian fibration f : M B with → a Lagrangian section Σ M, there exists a holomorphic coordinate system (z1,...,zn) on B such that ⊂
(i) zn = 0 is a local defining equation of D; (ii) on B D, dz1,...,dzn 1, dzn belong to Γ′ in the notation of Proposition 2.5; \ − (iii) there exists a symplectic basis p1,s,...,pn,s,q1,s,...,q1,n on each Λs = Γs,s B D satisfying { } ∈ \ p1,s = (dz1)s,...,pn,s = (dzn)s and the associated period matrix in the sense of Proposition 3.14 is given by 2 j ∂ Ψ ℓ θi = + log zn ∂zi∂zj 2π√ 1 − for some holomorphic function Ψ on B, which we call a potential function of the Lagrangian fibration, and some integer ℓ.
Proof. Let p1,...,pn,q1,...,qn 1 be as in Proposition 3.10 and Proposition 3.13. At a { − } point s B D, we add qn,s to get a symplectic basis of Λs. By analytic continuation, we get a multi-valued∈ \ 1-form q over B D such that any choice of a value q of q at a point n \ n,t n t B D, together with p1,t,...,pn,t,q1,t,...,qn 1,t, gives a symplectic basis of Λt. Using the∈ coordinate\ system in Proposition 3.13, we can− write n j qi = θi dzj , Xj=1 j where θi is a (univalent) holomorphic function on B for each 1 i n 1 and 1 j n, j ≤ ≤ − ≤ ≤ while θn is a multi-valued holomorphic function on B D for each 1 j n. By Proposition i j j\ ≤ ≤ 3.14, θ = θ for each 1 i, j n. It follows that θn is univalent holomorphic function on j i ≤ ≤ B for each 1 j n 1. By the choice of pn,s Ξs and Proposition 3.7, the monodromy operator τ : Λ≤ ≤Λ −is of the form ∈ s s → s τs(qn,s)= qn,s + ℓpn,s for some integer ℓ. Thus n n ℓ θ˜ := θ log zn n n − 2π√ 1 − ˜j j ˜j is also univalent. Set θi := θi if (i, j) = (n,n). Then θi is a univalent holomorphic function on B for all values of 1 i, j n and6 ≤ ≤ n q = θ˜jdz for 1 i n 1 i i j ≤ ≤ − Xj=1
n j ℓ qn = θ˜ dzj + log zn dzn. n 2π√ 1 Xj=1 − LOCAL STRUCTURE OF PRINCIPALLY POLARIZED STABLE LAGRANGIAN FIBRATIONS 11
Since qi’s are closed 1-forms on B, we have ∂θ˜j ∂θ˜k ∂θ˜i i = i = j ∂zk ∂zj ∂zk for any 1 i, j, k n. By Poincar´e’s lemma, there exists a holomorphic function Ψ such that ≤ ≤ 2 ˜j ∂ Ψ θi = . ∂zi∂zj
4. Construction of principally polarized stable Lagrangian fibrations with given potential functions
In this section, for a sufficiently small n-dimensional polydisk B with coordinate (z1,...,zn), we shall construct a principally polarized stable Lagrangian fibration f : (M,ωM ) B with a given potential function Ψ(z). Our construction closely follows Nakamura’s toroidal→ con- struction [Na]. However, main differences are the following: (i) the base space B is of dimension n (rather than 1). (ii) the total space should be not only smooth but also symplectic. (I) Construction of a non-proper Lagrangian fibration M˜ B. For each integer k Z, let E be a copy of C C equipped→ with linear coordinates ∈ k × (xk,yk). We define a complex manifold E by identifying points in k ZEk by the following ∪ ∈ rule: a point (xk,yk) of Ek with xk = 0 and yk = 0 is identified with a point (xk+1,yk+1) of E with x =0 and y = 0,6 if and only6 if k+1 k+1 6 k+1 6 2 1 xk+1 = xkyk, and yk+1 = . xk On E, zn := xkyk is a well-defined holomorphic function independent of k and k+1 k wn := xk− yk− is a meromorphic function independent of k, with zeros and poles supported on
k Z(xkyk = 0). ∪ ∈ Moreover, the 2-forms dyk dxk glue together yielding a holomorphic symplectic form ωE on E, satisfying ∧ dwn ωE = dzn . ∧ wn Fix coordinates (z1,...,zn 1, w1, . . . , wn 1) − − n 1 n 1 n 1 n 1 on C − C − and regard them as functions on the open subset C − (C×) − defined by × × w1 = 0, . . . , wn 1 = 0. 6 − 6 Define n 1 n 1 X˜ := C − (C×) − E. × × On X˜, we have the holomorphic functions z1,...,zn, w1, . . . , wn 1 and the meromorphic − function wn. 12 JUN-MUK HWANG, KEIJI OGUISO
n 1 n Define a morphismp ˜ : X˜ C by (z ,...,z ). The fiber ofp ˜ over b with zn(b) = 0 is isomorphic to → 6 n 1 (C×) − C× × with coordinates (w1, . . . , wn 1, wn) and the fiber over b with zn(b) = 0 is isomorphic to − n 1 1 (C×) − k ZPk ×∪ ∈ 1 1 where Pk is a copy of the projective line P with affine coordinate yk. We have a holomorphic symplectic 2-form n 1 n − dw dw ω := dz i + ω = dz i X˜ i ∧ w E i ∧ w Xi=1 i Xi=1 i on X˜. From now, we regard X˜ as a symplectic manifold by this symplectic form. From the coordinate expression of ωX˜ andp ˜, it is immediate thatp ˜ is a non-proper Lagrangian fibration. We denote M˜ = X˜ Cn B × where B = (z1,...,zn 1, zn) zi <ǫ( i) { − | | | ∀ } and ǫ is a sufficiently small positive real number. We denote the natural projection M˜ B induced fromp ˜ by → f˜ : M˜ B . → ˜ ˜ Note that the restriction ωM˜ of ωX˜ is a symplectic 2-form on M and f is a non-proper Lagrangian fibration. (II) Group action of Γ= Zn on M˜ . Let Ψ(z , z , , z ) be a holomorphic function on B such that the imaginary part 1 2 · · · n Im θ˜(z) of the Hessian matrix ∂2Ψ θ˜(z) = ( ) ∂zi∂zj is positive definite and ℓ be a positive integer. We define the period matrix θ(z) by
log zn On 1 0 θ(z)= θ˜(z)+ − . 2π√ 1 0 ℓ − We will write ˜ ˜ ˜ Θ1(z) Θ2(z) θ(z)= ˜ t ˜n , Θ2(z) θn(z) where Θ˜ (z) is the (n 1) (n 1) matrix, Θ˜ (z) is the (n 1) 1 matrix, Θ˜ t (z) is the 1 − × − 2 − × 2 transpose of Θ˜ (z) and θ˜n(z) is 1 1 matrix. 2 n × Set Γ = Zn 1 Z. We define a group action of Γ on M˜ as follows. Let γ = (j, m) − ⊕ ∈ Γ. Then the action T : M˜ M˜ is defined in terms of the coordinate functions on γ → Cn 1 (C )n 1 E M˜ by − × × − × k ⊂ T ∗z = z for i = 1,...,n 1 γ i i − LOCAL STRUCTURE OF PRINCIPALLY POLARIZED STABLE LAGRANGIAN FIBRATIONS 13
n 1 bi t n 1 bi T ∗(Π − w ) = exp (2π√ 1(jΘ˜ (z)b + mΘ˜ (z)b)Π − w γ i=1 i − 1 2 i=1 i n 1 where b = (b ) − is (n 1) 1 matrix, and i i=1 − × ˜ ˜n 1 Tγ∗xk = (exp (2π√ 1(jΘ2(z)+ mθn(z)))− xk mℓ − − ˜ ˜n Tγ∗yk = exp (2π√ 1(jΘ2(z)+ mθn(z))yk mℓ . − − n It is immediate that T T ′ = T ′ . Then T Aut (X/˜ C ) and γ T defines an γ∗ γ∗ γ∗+γ γ ∈ 7→ γ injective group homomorphism from Γ to Aut (M/˜ Cn). Here Aut (M/˜ Cn) is the group of automorphisms of M˜ over Cn, i.e., the group of automorphisms g of M˜ such that f˜ g = f˜. ◦ Proposition 4.1. The action Γ on M˜ is properly discontinuous, free and symplectic, in the sense that T ω = ω for each γ Γ. γ∗ M˜ M˜ ∈ Proof. Freeness of the action is clear from the description of the action. The proof of proper discontinuity is essentially the same as the proof of [Na], Theorem 2.6. This can be also seen from the concrete description of fibers below in (III), at least fiberwisely. Let us show that the action is symplectic, i.e., ωM˜ = Tγ∗ωM˜ for each γ = (j, m). This is a new part not considered by [Na]. We have Tγ∗dzi = dzi, Tγ∗dwi = exp (2π√ 1fi(z))wi for n 1 n 1 − all i, where, in terms of the standard basis e − of C , h iii=1 − fi(z)= jΘ˜ 1(z)ei + mΘ˜ 2(z)ei for 1 i n 1 and ≤ ≤ − ˜ ˜n fn(z)= jΘ2(z)+ mθn(z) . Thus, for i with 1 i n, we have ≤ ≤ Tγ∗dzi = dzi ,
dwi Tγ∗ = Tγ∗(d log wi) wi
= d log(T ∗w )= d(2π√ 1f (z)) + d(log w ) γ i − i i n dwi ∂fi = + 2π√ 1 dzk . wi − ∂zk Xk=1 Using these identities, we can compute n dwi Tγ∗ωM˜ = Tγ∗(dzi) Tγ∗( ) ∧ wi Xk=1
n n ∂f √ i = ωM˜ 2π 1 dzk dzi − − ∂zk ∧ Xi=1 Xk=1 ∂f ∂f √ i k = ωM˜ 2π 1 ( )dzk dzi . − − ∂z − ∂zi ∧ 1 Xi On the other hand, by definition of fi(z) (1 i n) and definition of θ˜(z) from the potential function Ψ(z), we know that ≤ ≤ n 1 − ˜i ˜i fi(z)= jαθα(z)+ mθn αX=1 n 1 − ∂2Ψ ∂2Ψ = j + m . α ∂z ∂z ∂z ∂z αX=1 α k n k Since Ψ(z) is holomorphic, it follows that ∂ ∂2Ψ ∂3Ψ ∂ ∂2Ψ ( )= = ( ) . ∂zk ∂zα∂zi ∂zk∂zα∂zi ∂zi ∂zα∂zk Substituting this into the formula above, we obtain that Tγ∗ωM˜ = ωM˜ . (III) Group quotient of M˜ by Γ= Zn. Let M = M/˜ Γ. By Proposition 4.1, M is a smooth symplectic manifold with symplectic form ω induced by ω and M admits a fibration f : M B induced by f˜. We denote M M˜ → the (scheme theoretic) fiber f 1(b) over b B by M . Let us describe the fibers M . − ∈ b b (III-1) Smooth fibers Mb First consider the case where z (b) = 0, i.e., the case where M is smooth. We have n 6 b n 1 n M˜ = (C×) − (x ,y ) x y = b (C×) . b × { k k | k k n}≃ (w1,...,wn−1,wn) k n and wn = zn(b) yk. Let ei i=1 be the ordered standard basis of Γ. From the description in (II), the action of Γ ish giveni by: j T ∗ w = exp(2π√ 1θ (b))w ei j − i j n T ∗ w = exp(2π√ 1θ (b))w ei n − i n for 1 i n 1 with θj = θ˜j and ≤ ≤ − i i j T ∗ w = exp(2π√ 1θ (b))w en j − n j n ℓ n T ∗ w = exp(2π√ 1θ˜ (b))z (b) w = exp(2π√ 1θ (b))w . en n − n n n − n n Hence Mb = M˜ b/Γ is an n-dimensional principally polarized abelian variety of period θ(b), as desired. By the description of ωM , the fibers Mb, zn(b) = 0, are also Lagrangian submanifolds. 6 Proposition 4.2. For b B, z (b) = 0, choose the basis ∈ n 6 p1,b,...,pn,b q1,b,...,qn,b of H (M , Z) such that M˜ = Cn/ p n and 1 b b h j,bij=1 n j qi,b = θi (b)pj,b Xj=1 LOCAL STRUCTURE OF PRINCIPALLY POLARIZED STABLE LAGRANGIAN FIBRATIONS 15 for each i (1 i n). Let ≤ ≤ 1 n 1 n pb ,...,pb qb ,...,qb 1 be the dual basis of H (Mb, Z). Then the integral 2-form n L := pi qi b b ∧ b Xi=1 give a monodromy invariant principal polarization of M over B D where D = (z = 0). \ n Proof. When z (b) = 0, the fiber M˜ ofp ˜ : M˜ B is (C )n and this family has no n 6 b → × monodromy over B (zn = 0). Thus we can fix a basis p1,b,...,pn,b of H1(M˜b, Z) uniformly in b, z (b) = 0. \ n 6 To get a basis of H1(Mb, Z), we choose additional elements q1,b,...,qn,b Hx(Mb, Z) ˜ ∈ determined by the deck-transformation of Mb induced by the action Te1 ,...,Ten . From the description of Te∗i on wj, they satisfy the relation n j qi,b = θi (b)pj,b. Xj=1 We see that q1,b,...,qn 1,b are invariant under the monodromy, while qn,b qn,b + ℓpn,b under the monodromy of− the generator γ of π (B D), i.e., the circle around7→ discriminant 1 \ divisor zn = 0. The 2-form Lb is a principal polarization on Mb. It remains to show that Lb is invariant under the monodromy. By definition of θ(b), we compute that n 1 − γ∗(L )= γ∗( p q )+ γ∗(p q ) b i,b ∧ i,b n,b ∧ n,b Xi=1 n 1 − = pi qi + pn (qn ℓpn)= L . b ∧ b b ∧ b − b b Xi=1 This implies the invariance. Remark 4.3. As in [Na], one can also describe f˜ : M˜ B in terms of toric geometry. Following an argument similar to [Na], Section 4, it seems→ possible to give a relatively principally polarized divisor (the relative theta divsor) which is defined globally over B D. However, its closure is not necessarily f-ample even if total space is of dimension 4 (cases\ of stable principally polarized Lagrangian 4-folds). In fact, a failure of f-ampleness of the closure already happens when the fiber dimension 2 and the base dimension 1 as explicitly described in [Na] Section 4, Page 219. See also Remark 3.6. (III-2) Singular fibers Mb Next consider the singular fibers of f. They are Mb with zn(b) = 0. Recall from (I) ˜ n 1 n 1 1 that Mb is the product of (C×) − with coordinate (wi)i=1− and the infinite tree k ZPk of 1 ˜ ∪ ∈ projective lines Pk with affine coordinate yk, and Mb = Mb/Γ. Let us denote by (0)k, ( )k 1 1 ∞ ∈ Pk the two points on the projective line Pk such that (0)k is identified with ( )k 1 in the tree. ∞ − 16 JUN-MUK HWANG, KEIJI OGUISO From (II), the action of Γ is given by: j T ∗ w = exp(2π√ 1θ˜ (b))w ei j − i j n T ∗ y = exp(2π√ 1θ˜ (b))y ei k − i k for 1 i n 1 and ≤ ≤ − j T ∗ w = exp(2π√ 1θ˜ (b))w en j − n j n Te∗n yk = exp(2π√ 1θn(b))yk ℓ , n − − where ei i=1 is the ordered standard basis of Γ. Here we note that the last equality shows that theh monodromyi operation corresponds to the shift of the components of the infinite 1 ˜ tree k ZPk. Thus M/ n 1 M and M ′ such that Φ is equivariant with respect to them. These C − -actions are free and all orbits have dimension n 1. Since both M and M ′ have trivial canonical bundle, the exceptional loci of the bimeromorphic− map must be of codimension 2. Since they n 1 ≥ n 1 are invariant under the C − -actions, they must be union of finitely many orbits of C − - actions. But then each component of the exceptional loci in M must be transformed to a component of the exceptional loci in M ′ biholomorphically. This implies that there are no exceptional loci and Φ is biholomorphic. As a corollary of Theorem 4.4 and Proposition 4.5, we obtain Corollary 4.6. For any principally polarized stable Lagrangian fibration, the positive inte- i i ger ℓ in Theorem 3.15 is the number of components of the singular fiber and S1 = S2 for each| i| in the notation of Definition 2.1. 6 Theorem 3.15, Theorem 4.4 and Corollary 4.6 complete the proof of Theorem 1.1. 5. Periods and the characteristic cycles In this section, we will examine the relation between types of the characteristic cycles and the periods. For simplicity, we will restrict our discussion to the case of ℓ = 1. The generalization to arbitrary ℓ is straightforward. Explicit constructions (Constructions I -III) will be given when n = 2, i.e., constructions of 4-dimensional principally polarized stable Lagrangian fibrations. Construction I gives an explicit example in which the types of characteristic cycles change fiber by fiber. Construction II gives an explicit example in which the types of characteristic cycles are constant type In (n< ) and Construction III gives an explicit example in which the types of characteristic cycles∞ are constant type A . ∞ Proposition 5.1. Let f : M B be a 2n-dimensional principally polarized stable La- grangian fibration with potential→ function Ψ(z) and ℓ = 1. We denote the (univalent) ˜ ˜j n period matrix by θ(z) = (θi (z))i,j=1 and the multi-valued period matrix θ(z) of f as log zn On 1 0 θ(z)= θ˜(z)+ − . 2π√ 1 0 1 − For b B for which M is singular, define n(b) (1 n(b) ) to be the order of ∈ b ≤ ≤∞ j n 1 j n 1 (exp(2π√ 1θ˜ (b))) − mod (exp(2π√ 1θ˜ (b))) − 1 i n 1 − n j=1 h − i j=1 | ≤ ≤ − i in the multiplicative group n 1 j n 1 (C×) − / (exp(2π√ 1θ˜ (b))) − 1 i n 1 . h − i j=1 | ≤ ≤ − i Then the characteristic cycle of Mb is of type In(b). Remark 5.2. The description above is simpler when f : M B is a 4-dimensional → principally polarized stable Lagrangian fibration with potential function Ψ(z) = Ψ(z1, z2) and ℓ = 1, as follows. We shall use this description in Constructions (I)-(III) below. We write the (univalent) period matrix θ˜(z) and the multi-valued period matrix θ(z) of f as θ˜ (z) θ˜ (z) log z 0 0 θ˜(z)= 1 2 ,θ(z)= θ˜(z)+ 2 . θ˜2(z) θ˜3(z) 2π√ 1 0 1 − 18 JUN-MUK HWANG, KEIJI OGUISO For b = (b1, 0) B for which Mb is singular, the characteristic cycle of Mb is then of type I , where n(b∈) (1 n(b) ) is exactly the order of n(b) ≤ ≤∞ exp(2π√ 1θ˜ (b))mod exp(2π√ 1θ˜ (b)) − 2 h − 1 i in the multiplicative group C / exp(2π√ 1θ˜ (b)) . × h − 1 i ˜ n 1 1 Proof. In the description (III-2), Mb is the quotient of Mb = k Z(C×) − Pk with n 1 n ∪ ∈ × coordinates ((w ) − ,y ) (k Z) by the action of Γ = Z with ordered standard basis j j=1 k ∈ e1,...,en 1, en . h − i In terms of the standard basis, the action is given by: n 1 j n 1 T ∗ : (w ) − (exp(2π√ 1θ˜ (b))w ) − ei j j=1 7→ − i j j=1 n T ∗ : y exp(2π√ 1θ˜ (b))y ei k 7→ − i k for e = i (1 i n 1), and for e ≤ ≤ − n n 1 j n 1 T ∗ : (w ) − (exp(2π√ 1θ˜ (b))w ) − en j j=1 7→ − n j j=1 ˜n Te∗n : yk exp(2π√ 1θn(b))yk 1 . 7→ − − ˜ n 1 1 n 1 n 1 Thus Mb/ en is (C×) − P0 in which ((wj)j=1− , 0) and ((wj′ )j=1− , ) are identified exactly h i n×1 n 1 n 1 ∞ when the two points (wj)j=1− and (wj′ )j=1− of (C×) − are in the same orbit under the action of the cyclic subgroup j n 1 G(b) := (exp(2π√ 1θ˜ (b))) − h − n j=1 i n 1 n 1 1 n 1 of (C×) − . On the other hand, ((C×) − P0)/ ei i=1− is the normalization of Mb. Thus, n 1 1 n×1 h i Mb is obtained from ((C×) − P0)/ ei i=1− by identifying the two (n 1) dimensional n 1 × n 1h i n 1 n 1 − − complex tori ((C×) − )/ ei i=1− and ((C×) − 0 )/ ei i=1− ,by the action of G(b) × {∞} h i n 1 ×n { 1} h i above. Here, as a subgroup of (C×) − , the group ei i=1− is the multiplicative subgroup generated by the n 1 elements h i − j n 1 (exp(2π√ 1θ˜ (b))) − , 1 i n 1 . − i j=1 ≤ ≤ − This implies the result. Construction I. Under the notation of Section 5, we set ℓ = 1 and (z + 5√ 1)3 + (z + 5√ 1)3 + 3z2z + 3z z2 Ψ(z , z ) := 1 − 2 − 1 2 1 2 , 1 2 6 Then z + z + 5√ 1 z + z log z2 0 0 θ˜(z)= 1 2 − 1 2 ,θ(z)= θ˜(z)+ z1 + z2 z1 + z2 + 5√ 1 2π√ 1 0 1 − − and y + y + 5 y + y Im θ˜(z)= 1 2 1 2 . y1 + y2 y1 + y2 + 5 LOCAL STRUCTURE OF PRINCIPALLY POLARIZED STABLE LAGRANGIAN FIBRATIONS 19 Here and hereafter xi and yi are the real and imaginary part of zi respectively. Since t + 5 > 0 and (t + 5)2 t2 > 0 when 2 2π√ 1z1 2π√ 1(z1+5√ 1) e − mod e − − h i in the multiplicative group C / e2π√ 1(z1+5√ 1) . By abuse of language, we include N = × h − − i ∞ when the order is not finite. Then, the characteristic cycle of M(z1,0) is of type IN . Proposition 5.3. In Construction 1, the characteristic cycle on M(z1,0) is of Type Ik with k< if and only if ∞ z Q(√ 1) . 1 ∈ − So, the singular fibers of finite characteristic cycle Ik (k < ) and the singular fibers of infinite characteristic cycle I are both dense over the disciminant∞ set. Moreover, the characteristic cycle of M is∞ precisely of type I (k < ) for z = 1/k. So, the singular (z1,0) k ∞ 1 fibers with characteristic cycles of type Ik with any sufficiently large k appear in this family. Proof. By the definition of N = N(z), it follows that N < for M(z1,0) if and only if there are integers k> 0 and m such that ∞ 2π√ 1z1 k 2π√ 1(z1+5√ 1) k (e − ) = (e − − ) . The last condition is equivalent to kz m(z + 5√ 1) Z 1 − 1 − ∈ which is also equivalent to (k m)x Z and (k m)y 5m = 0 . − 1 ∈ − 1 − Note that k m = 0 in the last equivalent condition, as otherwise k = m = 0. It is immediate to− see that6 two integers k> 0 and m satisfying last equivalent condition exist if and only if z Q(√ 1). Since e2π√ 1(z1+5√ 1) > 1 for z < 1, whereas e2π√ 1/k = 1 1 ∈ − | − − | | 1| | − | for k Z, it follows that the order N(z) for z = 1/k is precisely the order of e2π√ 1/k in ∈ 1 − the multiplicative group C×. This implies the last statement. Construction II. Under the notation of Section 5, we set ℓ = 1 and √ 1(z2 + z2) z z Ψ(z , z ) := − 1 2 + 1 2 , 1 2 2 k where n is a positive integer. Then √ 1 1/k log z2 0 0 θ˜(z)= − ,θ(z)= θ˜(z)+ 1/k √ 1 2π√ 1 0 1 − − 20 JUN-MUK HWANG, KEIJI OGUISO and 1 0 Im θ˜(z)= . 0 1 The matrix Im θ˜(z) is positive definite. So, taking a smaller 2-dimensional polydisk B, we obtain 4-dimensional Lagrangian fibration f : M B, associated with the potential → function Ψ(z) and ℓ = 1 above. The discriminant set is z2 = 0. The order of 2π√ 1/k 2π e − mod e− h i is exactly k in the multiplicative group C / e 2π , where 2π = 2π√ 1 √ 1. Then, the × h − i − − · − characteristic cycle of M(z1,0) is of type Ik, and in particular, the type is constant. Construction III. Under the notation of Section 5, we set ℓ = 1 and √ 1(z2 + z2) Ψ(z , z ) := − 1 2 + αz z , 1 2 2 1 2 where α is any irrational, real number, say √2. Then √ 1 α log z2 0 0 θ˜(z)= − ,θ(z)= θ˜(z)+ α √ 1 2π√ 1 0 1 − − and 1 0 Im θ˜(z)= . 0 1 The matrix Im θ(z) is positive definite. So, taking a smaller 2-dimensional polydisk B, we obtain 4-dimensional Lagrangian fibration f : M B, associated with the potential → function Ψ(z) and ℓ = 1 above. The discriminant set is z2 = 0. 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[Vo] Voisin, C.: Hodge theory and complex algebraic geometry II, Cambridge University Press, 2003 Jun-Muk Hwang, Korea Institute for Advanced Study, Hoegiro 87, Seoul, 130-722, Korea E-mail address: [email protected] Keiji Oguiso, Department of Mathematics, Osaka University, Toyonaka 560-0043 Osaka, Japan and Korea Institute for Advanced Study, Hoegiro 87, Seoul, 130-722, Korea E-mail address: [email protected]